SECTION FORMULA

About "Section formula"

Section formula :

We use the section formula to find the point which divides the line segment in a given ratio.

The point P which divides the line segment joining the two points A (x1,  y1) and B (x2, y2) internally in the ratio l : m is  

If P divides a line segment AB joining the two points A (x1,  y1) and B (x2, y2) externally in the ratio l : m is,

Let us look into some example problems based on the above concepts.

Example 1 :

Find the point which divide the line segment joining the points (-1,2) and (4,-5) internally in the ratio 2 : 3. 

Solution :

Since we need to find the point which divides the line segment internally in the ratio 2 : 3, we have to use the formula

  =   (lx2 + mx1) / (l + m),   (ly2 + my1) / (l + m)

Here (x1, y1)  ==> (-1, 2) (x2, y2) ==>  (4, -5) and l : m ==> 2:3

  =  2(4) + 3(-1) / (2 + 3)  ,  2(-5) + 3(2) / (2 + 3)

  =  [(8 - 3) / 5 ,  (-10 + 6) / 5]

  =  (5 / 5 ,  -4 / 5)

  =  (1, -4/5)

Hence the point (1, -4/5) divides the line segment internally in the ratio 2 : 3.

Example 2 :

Find the point which divide the line segment joining the points (2, 1) and (3, 5) externally in the ratio 2 : 3. 

Solution :

Since we need to find the point which divides the line segment externally in the ratio 2 : 3, we have to use the formula

  =   (lx2 - mx1) / (l - m),   (ly2 - my1) / (l - m)

Here (x1, y1)  ==> (2, 1) (x2, y2) ==>  (3, 5) and l : m ==> 2:3

  =  2(3) - 3(2) / (2 - 3)  ,  2(5) - 3(1) / (2 - 3)

  =  [(6 - 6) / 5 ,  (10 - 3) / 5]

  =  (0 / 5 ,  7 / 5)

  =  (0, 7/5) 

Hence the point (0, 7/5) divides the line segment externally in the ratio 2 : 3.

Example 3 :

Find the ratio in which x axis divides the line segment joining the points (6 , 4) and (1 ,- 7).

Solution :

Let l : m be the ratio of the line segment joining the points (6,4) and (1,-7) and let p(x,0) be the point on the x axis

Section formula internally

=  (lx₂ + mx₁)/(l + m) , (ly₂ + my₁)/(l + m)

(x, 0)  =  [(1) + m(6)]/(l + m) , [l(-7) + m(4)]/(l + m)

(x , 0)  =  [l + 6 m]/(l + m) , [-7l + 4m]/(l + m)

Equating y-coordinates

[-7l + 4m]/(l + m)  =  0

  - 7 l + 4 m  =  0

  - 7 l  =  - 4 m

  l/m  =  4/7

  l : m  =  4 : 7              

Hence x-axis divides the line segment in the ratio 4 : 7. 

After having gone through the stuff given above, we hope that the students would have understood "Section Formula ". 

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